Compact hyperbolic Coxeter n-polytopes with n+3 facets

We use methods of combinatorics of polytopes together with geometrical and computational ones to obtain the complete list of compact hyperbolic Coxeter n-polytopes with n+3 facets, 3<n<8. Combined with results of Esselmann (1994), Andreev (1970) and Poincare (1882) this gives the classification of all compact hyperbolic Coxeter n-polytopes with n+3 facets.Comment: v4: paper is rewritten. Complete proofs added, errors corrected. 36 pages, a lot of figure

Partial mirror symmetry, lattice presentations and algebraic monoids

This is the second in a series of papers that develops the theory of reflection monoids, motivated by the theory of reflection groups. Reflection monoids were first introduced in arXiv:0812.2789. In this paper we study their presentations as abstract monoids. Along the way we also find general presentations for certain join-semilattices (as monoids under join) which we interpret for two special classes of examples: the face lattices of convex polytopes and the geometric lattices, particularly the intersection lattices of hyperplane arrangements. Another spin-off is a general presentation for the Renner monoid of an algebraic monoid, which we illustrate in the special case of the "classical" algebraic monoids.Comment: 41 page

New hyperbolic 4-manifolds of low volume

We prove that there are at least 2 commensurability classes of minimal-volume hyperbolic 4-manifolds. Moreover, by applying a well-known technique due to Gromov and Piatetski-Shapiro, we build the smallest known non-arithmetic hyperbolic 4-manifold.Comment: 21 pages, 6 figures. Added the Coxeter diagrams of the commensurability classes of the manifolds. New and better proof of Lemma 2.2. Modified statements and proofs of the main theorems: now there are two commensurabilty classes of minimal volume manifolds. Typos correcte

Subword complexes and nil-Hecke moves

For a finite Coxeter group W, a subword complex is a simplicial complex associated with a pair (Q, \rho), where Q is a word in the alphabet of simple reflections, \rho is a group element. We describe the transformations of such a complex induced by nil-moves and inverse operations on Q in the nil-Hecke monoid corresponding to W. If the complex is polytopal, we also describe such transformations for the dual polytope. For W simply-laced, these descriptions and results of \cite{Go} provide an algorithm for the construction of the subword complex corresponding to (Q, \rho) from the one corresponding to (\delta(Q), \rho), for any sequence of elementary moves reducing the word Q to its Demazure product \delta(Q). The former complex is spherical if and only if the latter one is the (-1)-sphere.Comment: 6 pages. Comments welcome! arXiv admin note: substantial text overlap with arXiv:1305.5499; and text overlap with arXiv:1111.3349 by other author